Anatomy - Inria · 3 X. Pennec Computational Anatomy 5 Modeling and image analysis: a virtuous loop Normalization, Interpretation, Modeling Identification Personalization

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X. Pennec Computational Anatomy 1

Xavier Pennec

Statistical Computing on Manifolds for Computational Anatomy

Statistical registration: Pair-wise and Group-wise

Alignment and Atlas formation,

Brisbane, Australia, Friday Nov 2, 2007.


Anatomy

Gall (1758-1828) : PhrenologieTalairach (1911-2007)

Antiquity • Animal models • Philossphical physiology

Renaissance:• Dissection, surgery• Descriptive anatomy

Vésale (1514-1564)Paré (1509-1590)

1990-2000:• Explosion of imaging • Computer atlases• Brain decade

2007

Science that studies the structure and the relationship in space of different organs and tissues in living systems

[Hachette Dictionary]

Revolution of observation means (1988-2007) : From dissection to in-vivo in-situ imaging

Galien (131-201)

1er cerebral atlas, Vesale, 1543

17-20e century:• Anatomo-physiology• Microscopy, histology

Visible Human Project, NLM, 1996-2000Voxel-Man, U. Hambourg, 2001

Talairach & Tournoux, 1988

Sylvius (1614-1672)Willis (1621-1675)

Paré, 1585

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The revolution of medical imaging

In vivo observation of living systems A large number of modalities to image

anatomy and function Growing spatial resolution (molecules to whole body)

Multiple temporal scales

Non invasive observations Emergence of large databases

From representative individual to population

Extract and structure information 50 to 150 MB for a clinical MRI

Computer analysis is necessary

From descriptive atlases to interactive and generative models (simulation)

MNI 305 (1993), ICBM 152 (2001), Brain web (1999) Bone Morphing® (Fleute et al, 2001)

200 µm

Mouse colonMicro-vaissels

brain

850 µmheart


Algorithms to Model and Analyze the Anatomy Estimate representative organ anatomies across species, populations, diseases, aging, ages… Model organ development across time Establish normal variability

To understand and to model how life is functioning Classify pathologies from structural deviations (taxonomy) Integrate individual measures at the population level to relate anatomy and function

To detect, understand and correct dysfunctions From generic (atlas-based) to patients-specific models Quantitative and objective measures for diagnosis Help therapy planning (before), control (during) and follow-up (after)

Computational Anatomy

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Modeling and image analysis: a virtuous loop

Normalization,Interpretation,

Modeling

IdentificationPersonalization

AnatomyAnatomyAnatomyAnatomyPhysicsPhysicsPhysicsPhysicsPhysiologyPhysiologyPhysiologyPhysiology

Images,Signals,Clinics,Genetics,etc.

Individual

Computational models of the human body

PopulationPrevention

Diagnosis

Therapy

Computer assisted medicine

Generative models

Statistical analysis

Knowledge inference

Integrative modelsfor biology and neurosciences


Methods of computational anatomy

Hierarchy of anatomical manifolds (structural model s) Landmarks [0D]: AC, PC [Talairach et Tournoux, Bookstein], functional landmarks Curves [1D]: crest lines, sulcal lines [Mangin, Barillot, Fillard…] Surfaces [2D]: cortex, sulcal ribbons [Thompson, Mangin, Miller…], Images [3D functions]: VBM, Diffusion imaging Transformations: rigid, multi-affine, local deformations (TBM), diffeomorphisms

[Asburner, Arsigny, Miller, Trouve, Younes…]

Groupwise correspondances in the population

Model observations and its structural variability

Statistical computing on Riemannian manifolds

Structural variability of the cortex

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Outline

Goals and methods of Computational anatomy

Statistical computing on manifolds The geometrical and statistical framework Examples with rigid body transformations and tensors

Morphometry of the Brain Statistics on curves to model the cortex variability Local statistics on local deformations Towards global statistics on (some) diffeomorphisms

Challenges of Computational Anatomy


Exponential chart (Normal coord. syst.) : Development in tangent space along geodesics

Geodesics = straight lines

The geometric framework: Riemannian ManifoldsRiemannian metric :

Dot product on tangent space Speed, length of a curve

Distance and geodesics Closed form for simple metrics/manifolds Optimization for more complex

Gradient descent

Distance

Addition

Subtraction

Riemannian manifoldEuclidean spaceOperator

)( ttt xCxx ∇−=+ εε

)(log yxy x=xyxy +=

xyyx −=),(distx

xyyx =),(dist

)(exp xyy x=

))((exp txt xCxt

∇−=+ εε

xyxy −=

Unfolding (log x), folding (exp x)

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Statistical computing on manifolds

First statistical tools Fréchet / Karcher mean: minimize variance

Covariance Higher order moments in the exponential chart

Intrinsic vs. extrinsic [Oller & Corcuera 95, Battacharya & Patrangenaru 2002 ]

Distributions and tests : practical approximations

Gaussian maximizes entropy knowing mean and covariance

Mahalanobis distance follows a chi2 law

Intrinsic Riemannian computing Interpolation (bi- tri-linear), filtering (e.g. Gaussian): weighted means

Harmonic / anisotropic regularization: Laplace-Beltrami

Trivial numerical scheme in the exponential chart

[ Pennec, NSIP’99, JMIV 25(1) 2006, Pennec et al, IJCV 66(1) 2006]

( ) ( ) ( ) ( )rOkN / Ric avec 2/yx..yxexp.)y(3

1)1(T

σεσ ++−=

= −

ΣΓΓ

( )rOn

t

/)( x..x)( 32)1(2 σεσχµ ++Σ= −~xxx xxx


Statistical Analysis of the Scoliotic Spine

Database 307 Scoliotic patients from the Montreal’s

Sainte-Justine Hospital. 3D Geometry from multi-planar X-rays

Mean Main translation variability is axial (growth?) Main rotation var. around anterior-posterior axis

PCA of the Covariance 4 first variation modes have clinical meaning

[ J. Boisvert et al., ISBI’06, to appear in IEEE TMI , 2007]

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Statistical Analysis of the Scoliotic Spine

• Mode 1: King’s class I or III

• Mode 2: King’s class I, II, III

• Mode 3: King’s class IV + V

• Mode 4: King’s class V (+II)

[ J. Boisvert et al., ISBI’06, to appear in IEEE TMI , 2007]


Liver puncture guidance using augmented reality3D (CT) / 2D (Video) registration

2D-3D EM-ICP on fiducial markers

Certified accuracy in real time

Validation Bronze standard (no gold-standard)

Phantom in the operating room (2 mm) 10 Patient (passive mode): < 5mm (apnea)

[ S. Nicolau, PhD’04 MICCAI05, Comp. Anim. & Virtua l World 2005, MICCAI’07 ]

S. Nicolau, IRCAD / INRIAS. Nicolau, IRCAD / INRIA

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Affine invariant metric (curved space – Hadamard)

Geodesics

Distance

Log-Euclidean metric (vector space)

Geodesics

Distance

Metrics for Tensor computing

[ Arsigny, Pennec, Fillard, Ayache: Fast and Simple Calculus on Tensors in the Log-Euclidean Framework, MICCAI’05, SIMAX 29(1) 2007, MRM 52(6) 2006 ]

2/12/12/12/1 )..exp()(exp ΣΣΣΨΣΣ=ΣΨ −−Σ

22/12/12

2

)..log(|),(distL

−−

ΣΣΨΣ=ΣΨΣΨ=ΨΣ

( ) ( ) ( ) 2

21

2

21 loglog,dist Σ−Σ≡ΣΣ

)exp()(exp ΣΨ=ΣΨΣ

[ Pennec, Fillard, Ayache: A Riemannian Framework for Tensor Computing, IJCV 66(1), 2006. ]

[Fletcher 2004, Moakher SIAM-X 2005, Lenglet, JMIV 2006]


Filtering and anisotropic regularization of DTIRaw Coefficients σσσσ=2

Riemann Gaussian σσσσ=2 Riemann anisotropic

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DTI-based Anatomical models

Diffusion tensor IRMCovariance of the water Brownian motion

Estimation, filtering, interpolation

[ J.M. Peyrat et al, MICCAI’06,To appear in TMI, 2008]

Atlas of the heart fibers 7 DTI of dogs hearts

Fibers and sheets structure

Fiber extraction: architecture of axons tracts

[Pennec et al, IJCV 66(1) 2006, Fillard et al, ISBI’06 and IEEE TMI, in press]


MedINRIA

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Outline

Goals and methods of Computational anatomy

Statistical computing on manifolds The geometrical and statistical framework Examples with rigid body transformations and tensors

Morphometry of the Brain Statistics on curves to model the cortex variability Local statistics on local deformations Towards global statistics on (some) diffeomorphisms



Morphometry of the Cortex from Sucal Lines

[ Fillard et al., IPMI 2005, Neuroimage 34(2), 2007, Extension later today!]

2/ Computation of the covariance tensor at each point of the mean

80 instances of 72 sulci

∫ ∫∑Ω Ω

Σ=

Σ∇+ΣΣ−=Σ 2

)(1

2

2)),(()(

2

1)(

x

n

i

ii dxxdistxxGCλ

σ

1/ Mean curve: Alternated minimization of the variance on matches / position

3/ Extrapolation to the whole volume :harmonic diffusion of tensors

Silvian fissure

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Comparison with a diffeomorphic approach

Difference between Matching, then extrapolation

[Fillard, NeuroImage 34(2) 2007]

Extrapolation of speed vectors and trajectory integration

Method Global space diffeomorphism

parameterized by a finite number of point

Distance between lines using currents[J. Glaunès, M. Vaillant: IPMI 2005]

Advantages Generative model of deformations

Retrieve the tangential deformation component.

[ S, Durrleman, X. Pennec, A. Trouvé, N. Ayache, MICC AI 2007 ]

Variability at each point

Global variability (2 PGA modes)


Variability for what?

Several methods with different assumptions: Similar results at some locations, different results at other places Vary assumptions and discover truth by consensus

Learning / modeling phase (anatomy / neurosciences) Goal: analyze and understand the population variability Identify anatomical differences between populations

Can be computationally intensive, Relies on good quality observations and “mild pathologies”

Use in a clinical / medical workflow : Personalization of atlases Anatomical prior to compensate for incomplete / noisy / abnormal

(pathological) observations. How to use statistics as a regularizer in registration? Need robust methods, should be very fast (at least efficient)

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One example use of variability information: better constrain the atlas to subject registration

Deform the atlas anatomy (without tumor) towards the patient one Segment the structures of interest in the patient image Minimize irradiation in areas at risk.

[ Commowick, et al, MICCAI 2005]


Introducing local variability and pathologies in non-linear registration

Non stationary regularization: anatomical prior on the deformability Non stationary image similarity / regularization tradeoff:

takes pathologies into account

[ Runa. Stefanescu et al, PPL 14(2), 2004 & Med. Im age Analysis 8(3), 2003]

Patient Stiffness field Atlas to patient With pathology

)(.))(,()( TETJIETE regsim λ+=

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Regularization in dense non-linear registration

Physically based regularizations

Elastic [Bajcsy 89] Fluid [Christensen TMI 97]

Right-invariant distance [LDDMM, Beg IJCV 05]

Efficient regularization methods

Gaussian filtering [Thirion Media 98, Modersitzki 2004]

Isotropic but non stationary [Lester IPMI’99]

Towards anisotropic non stationary regularization [Stefanescu MedIA 2004]

Observation:

No regularization model is more justified than others

Learning statistically the variability from a population[Thompson 2000, Rueckert TMI 2003, Fillard IPMI 2005]


Statistics on the deformation field• Objective: planning of conformal brain radiotherapy (O. Commowick, Dosisoft)

• 30 patients, 2 to 5 time points (P-Y Bondiau, MD, CAL, Nice)

[ Commowick, et al, MICCAI 2005, T2, p. 927-931]

Robust

∑ Φ∇=i iN

xxDef )))((log(abs)(1

∑ Σ=∑i iN

xx )))((log(abs)(1

1))(()( −∑+= xIdxD λ

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Introducing deformation statistics into RUNA

Scalar statistical stiffness Tensor stat. stiffness (FA)Heuristic RUNA stiffness

[ Commowick, et al, MICCAI 2005, T2, p. 927-931]


Riemannian elasticity : a well posed framework to introduce statistics in non-linear elastic regularization

Gradient descent

Including statistics in Regularization

St Venant Kirchoff elastic energy Elasticity is not symmetric Statistics are not easy to include

Idea: Replace the Euclidean by the Log-Euclidean metric

Statistics on strain tensors: Mean, covariance, Mahalanobis computed in Log-space

Using Riemannian Elasticity as a metric (TBM) The mean provides an unbiased atlas

Better constraining the deformation should give better results in TBM

)(Reg),Images(Sim)( Φ+Φ=ΦC

Φ∇Φ∇=Σ .t

( ) ( ) ( )22 Tr2

)(Tr Reg II −Σ+−Σ=Φ ∫λµ

[ Pennec, et al, MICCAI 2005, LNCS 3750:943-950, MF CA’06]

( ) 222 )log(),(dist )(Tr Σ=Σ→−Σ II LE

( ) ( ) ( )∫ −Σ−Σ=Φ − WWgT

)log(Vect.Cov.)log(VectRe 1

[ Natsha Lepore + Caroline Brun + Paul Thompson, Equipe Associee Brain Atlas with UCLA]

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Statistics on which deformations feature?Local statistics on local deformation (mechanical p roperties)

Gradient of transformation, strain tensor

[Riemannian elasticity, TBM, N. Lepore + C. Brun]

Global statistics on displacement field or B-spline parameters [Rueckert et al., TMI, 03], [Charpiat et al., ICCV’05], [P. Fillard, stats on sulcal lines]

Simple vector statistics, but inconsistency with group properties

Space of “initial momentum” [Quantity of motion inste ad of speed] [Vaillant et al., NeuroImage, 04] [Durrleman et al, MICCAI’07]

Based on left-invariant metrics on diffeos [Trouvé, Younes et al.]

Needs theoretically a finite number of point measures

Computationally intensive

An alternative: log-Euclidean statistics on diffeom orphisms?

[Bossa, MICCAI’07, Vercauteren MICCAI’07, Ashburner NeuroImage 2007]

Mathematical problems but efficient numerical methods!


Statistics on geometrical objects

A consistent statistical computing framework on Riemannian manif olds with important applications in

Medical Image Analysis (registration evaluation, DTI) Building models of living systems (spine, brain, heart…)

Is the Riemannian metric the minimal structure?

No bi-invariant metric but bi-invariant means on Lie groups [V. Arsigny] Change the Riemannian metric for the symmetric Cartan connection?

Infinite dimensional manifolds Curves and surfaces Space of diffeomorphisms

How to chose or estimate the metric? Invariance, reacheability of boundaries, learning the metric Families of anatomical deformation metrics (models of the Green’s function)

Spatial correlation between neighbors… and distant points

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Build models from multiple sources Curves, surfaces [cortex, sulcal ribbons]

Volume variability [Voxel/Tensor Based Morphometry, Riemannian elasticity]

Diffusion tensor imaging [fibers, tracts, atlas]

Topological changes

Evolution: growth, pathologies

Compare and combine statistics on anatomical manifolds Each method is biased by its assumptions (fewer data than unknowns)

Validate methods and models by consensus Integrative model (transformations ?)

Couple modeling and statistical learning Variability estimation / structure inference / model validation and refinement

Use models as a prior for inter-subject registration / segmentation:Validation by better statistics on populations (e.g. functional)

Need large database and distributed processing/algorithms


Acknowledgements

Associated team Brain Atlas with LONI: P. Thompson, N. Lepore, C. Brun

Brain variability: P. Fillard, V. Arsigny, S. Durrleman,

Spine shape: J. Boisvert, F. Cheriet, Ste Justine Hospital, Montreal.

Inter-subject non-linear registration: P. Cathier, R. Stefanescu, O. Commowick

ACI Agir / Grid computing: T. Glatard and J. Montagnat, I3S.

N. Ayache and current / former Epidaure / Asclepios team members.

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Anatomy - Inria · 3 X. Pennec Computational Anatomy 5 Modeling and image analysis: a virtuous loop Normalization, Interpretation, Modeling Identification Personalization